Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.
problem Properties of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
method Construction and study of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
result Hypoelliptic logarithmic Sobolev inequalities on the space.
Researchers study ancient and finite solutions of Laplacian coflow and modified coflow on a 7D Heisenberg group.
problem Analyzing the behavior of G2-structures under Laplacian and modified Laplacian coflow on a 7D Heisenberg group. method Examining the ancient, finite, and eternal solutions of the Laplacian coflow and modified coflow.
result Ancient solutions for Laplacian coflow and finite, ancient, or eternal solutions for modified coflow on the 7D Heisenberg group.
Study joint spectrum of sub-Laplacian and Reeb vector field on Heisenberg Bieberbach manifolds.
problem Computing the joint spectrum of sub-Laplacian and Reeb vector field on specific manifolds.
method Explicit computation using theta functions and character theory of finite groups.
result Explicitly computed multiplicities of the joint spectrum.
Study finds minimal length networks connecting three points in Heisenberg group.
problem Finding minimal length networks connecting three points in the Heisenberg group.
method Proved existence of minimal horizontal triods, formulated curve shortening flow, used numerical experiments.
result Characterized and deformed minimal horizontal triods into critical points for length functional.
Study Heisenberg homology on surface configurations, revealing new representations of mapping class groups.
problem Homology of surface configurations with Heisenberg group representations.
method Analysis of unordered configurations in a surface, using Heisenberg group actions and representations.
result Obtained genuine and projective representations of mapping class groups from Heisenberg group actions.
The paper studies groups formed by two parabolic maps and their properties.
problem Understanding groups generated by two parabolic maps in mSU(2,1). method Analyzes conditions for the group to be discrete and free, and calculates the diameter of a circle in the Heisenberg group.
result Conditions are provided to ensure the group is discrete and free.
New examples of double Kodaira fibrations found using surface braid groups.
problem Finding new examples of double Kodaira fibrations.
method Using finite Galois covers of a product of curves and explicit group epimorphisms from surface braid groups to finite Heisenberg groups.
result Every curve of genus b is the base of a double Kodaira fibration; the number of non-isomorphic surfaces is at least $oldsymbolω(b+1)$. In this paper, on the first, we prove Δr=2H where Δ is the Laplacian operator, r=(r1,r2,r3) the position vector field and H is the mean curvature vector field of a surface S in the 3-dimensional Heisenberg group H3. In the second, we classify the ruled surfaces by straight…
This paper defines and studies currents and slices in the Heisenberg group, with new challenges and insights.
problem Defining and studying currents and slices in the Heisenberg group Hn. method Definition and classification of currents, slicing of currents, and analysis of properties.
result New challenges and insights in the study of currents on the Heisenberg group, including a unique slice dimension.
Examples of area-minimizing graphs with low regularity in a specific group.
problem Finding area-minimizing graphs with low regularity in a sub-Finsler Heisenberg group.
method Providing examples of entire area-minimizing horizontal graphs with prescribed singular sets.
result Examples of area-minimizing graphs that are locally Lipschitz but not necessarily smoother.
We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give Lp-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvatu…
New manifold groups contain all special and extra-special p-groups.
problem Understanding the diffeomorphism groups of compact manifolds.
method Constructing new compact manifolds and analyzing their diffeomorphism groups.
result Contains all extra-special and special p-groups of order pr. New method polarizes anisotropic Heisenberg groups.
problem Polarizing anisotropic Heisenberg groups.
method Implementing a technique to polarize anisotropic Heisenberg groups.
result New class of polarizable Carnot groups expanded.
We give new examples of entire area-minimizing t-graphs in the subriemannian Heisenberg group H^1. Most of the examples are locally lipschitz in Euclidean sense. Some regular examples have prescribed singular set consisting of either a horizontal line or a finite number of horizontal halflines extending from a given po…
Invariant of 3-manifolds using Hopf algebra elements.
problem Constructing an invariant for closed 3-manifolds.
method Using involutory Hopf algebra elements and normal o-graphs.
result Invariant values in cyclic quotient of Heisenberg double.
Study Ricci-Bourguignon flow on Heisenberg and quaternion Lie groups.
problem Deformation of spectrum and length spectrum on compact nilmanifolds.
method Ricci-Bourguignon flow on Heisenberg and quaternion Lie groups.
result Construct a solution of the Ricci-Bourguignon flow on Heisenberg and quaternion nilpotent Lie groups.
New conformally Einstein metrics on Heisenberg group found.
problem Finding new conformally Einstein metrics on specific Lie groups.
method Described Lorentzian semi-direct extensions of the Heisenberg group.
result Classified Bach-flat left-invariant Lorentzian metrics.
Study examines geometry of special metric on Heisenberg group.
problem Geometry of left-invariant Randers metrics on the Heisenberg group.
method Investigation of left-invariant Randers metrics on the Heisenberg group.
result Analysis of the geometry of these metrics.
Study submanifolds with boundary in Heisenberg groups, proving Stokes' Theorem.
problem Understanding submanifolds with boundary in sub-Riemannian Heisenberg groups.
method Introduced examples and proved Stokes' Theorem involving Rumin's differential forms.
result Stokes' Theorem for submanifolds with boundary in Heisenberg groups.
Classifies geodetically convex sets and functions on Heisenberg group.
problem Characterizing geodetically convex sets and functions in the Heisenberg group.
method Classification through mathematical analysis.
result Geodetically convex sets and functions defined on Heisenberg group Hn classified. Study lifts plane mappings to Heisenberg group.
problem Contact quasiconformal mappings in hyperbolic Heisenberg group.
method Lifting Theorem for symplectic mappings.
result Symplectic mappings lifted to Heisenberg group.
Homogeneous magnetic paths found in Heisenberg space.
problem Understanding magnetic geodesics in the Heisenberg group.
method Proving homogeneity of magnetic geodesics derived from the canonical contact structure.
result Magnetic geodesics in the Heisenberg group are homogeneous.
Let D be a closed disk centered at the origin in the horizontal hyperplane {t=0} of the sub-Riemannian Heisenberg group $\hh^n$, and C the vertical cylinder over D. We prove that any finite perimeter set E such that D⊂E⊂C has perimeter larger than or equal to the one of the rotationally symm…
Study submanifolds with boundaries in Heisenberg groups using Stokes' Theorem.
problem Understanding submanifolds with boundaries in Heisenberg groups.
method Introduce and study CH1-regular submanifolds with boundaries, prove Stokes' Theorem for them. result Prove a version of Stokes' Theorem for submanifolds with boundaries in Heisenberg groups.
Find conditions for starshapedness of level sets in Heisenberg group.
problem Ensure starshapedness of level sets of p-capacitary potentials. method Examine horizontally p-harmonic functions in the Heisenberg group. result Sharp conditions for strictly starshaped level sets.
Defines contact structures on Heisenberg groups for geometric interpretation.
problem Finding a geometric interpretation for the Yamabe equation on Heisenberg groups.
method Defines contact structures of Heisenberg type, introduces a natural connection, and computes conformal scalar curvature.
result Establishes equivalence between contact Riemannian manifolds and contact structures of Heisenberg type.
Computes minimal polynomials for generalized Heisenberg groups.
problem None explicitly stated; focus on method.
method Computes minimal polynomials for generalized Heisenberg groups.
result Explicit minimal polynomials for generalized Heisenberg groups.
The paper describes geodesics on a Kähler cone of the Heisenberg group.
problem Understanding geodesics on the Kähler cone of the Heisenberg group.
method Analyzing the Heisenberg group to describe geodesics.
result It is not a complete manifold.
Configurations on Heisenberg group are flat surfaces.
problem Characterizing equidistant triples on the Heisenberg group.
method Proved isomorphism to a hypersurface in R^3.
result Configuration space is a flat surface.
The paper constructs TQFTs and Schrödinger representations for Heisenberg group.
problem Constructing TQFTs and Schrödinger representations for Heisenberg group.
method Using Lagrangian correspondences and q-deformation of U(1).
result Normalization of Schrödinger bimodule action reproduces abelian TQFT.
Researchers create non-homogeneous finite-volume ends on quaternionic Kähler manifolds.
problem Constructing non-homogeneous quaternionic Kähler manifolds with finite volume ends.
method Using cohomogeneity one deformation of symmetric spaces, the researchers constructed manifolds with specific fundamental groups.
result The constructed manifolds are aspherical and have finite volume ends, not locally homogeneous.
In Heisenberg group, bisectors are spinal spheres with specific curvature.
problem Understanding bisectors in the Heisenberg group.
method Showed bisectors are spinal spheres and calculated their curvature.
result Metric bisectors in Heisenberg group are spinal spheres with specific curvature.
Survey on non-characteristic domains in Heisenberg groups.
problem Characterizing non-characteristic domains in Heisenberg groups.
method Analyzing diffeomorphisms and proving conjectures.
result Bounded domains diffeomorphic to solid torus are non-characteristic.
Criterion for surfaces in Heisenberg group to be graphs using flat cones.
problem Characterizing surfaces in Heisenberg group as graphs.
method Using planar cones to define intrinsic rectifiability.
result Criterion for topological surfaces to be intrinsic Lipschitz graphs.
Averages vanish in ℓ1 cohomology of Heisenberg groups.
problem Vanishing of averages in ℓ1 cohomology. method Proof of vanishing averages for abelian and Heisenberg groups.
result Vanishing of ℓ1 cohomology for Heisenberg groups. Translation surfaces in Heisenberg group classified by Gauss map determinant.
problem Classifying translation surfaces with zero intrinsic curvature in Heisenberg group.
method Constructed surfaces as product of planar curves, classified by Gauss map determinant.
result Surfaces with vanishing intrinsic curvature identified.
The paper proves uniqueness and existence of pseudohermitian submanifolds in Heisenberg groups.
problem Geometric properties of pseudohermitian submanifolds in Heisenberg groups.
method Study and prove uniqueness and existence theorems.
result Uniqueness and existence theorems for pseudohermitian submanifolds in Heisenberg groups.
New categorical actions link topological and algebraic structures.
problem Understanding relationships between topological and algebraic structures.
method Categorical actions of type B braid group on homotopy categories.
result Proves Rouquier's conjecture on faithfulness of Type B 2-braid group.
Study maps surface configurations to Heisenberg homologies for mapping class groups.
problem Understanding Mapping Class Groups of punctured surfaces.
method Action of mapping classes on Heisenberg homologies of surface configurations.
result Representations of Mapping Class Groups derived from Heisenberg homologies.
We show certain symmetry of the dimensions of cohomologies of the funda- mental groups of compact Sasakian manifolds by using the Hodge theory of twisted basic cohomology. As applications, we show that the polycyclic fundamental groups of compact Sasakian manifolds are virtually nilpotent and Sasakian solvmanifolds are…
Study shows only hyperplanes in Heisenberg groups have zero curvature.
problem Understanding Bernstein problem in higher dimensional Heisenberg groups.
method Sub-Riemannian characterization of ruling property and study of geodesics.
result Only hyperplanes have zero horizontal symmetric second fundamental form in Heisenberg groups.
Among eight possible geometric structures on three-dimensional manifolds less studied from the differential geometric point of view are those modelled on the Heisenberg group Heis3. We consider the Heisenberg left-invariant metric and use some results on Levi-Civita connection and curvature tensor to present solutio…
Develops analysis of Hölder continuous mappings on Heisenberg groups.
problem Analyzing Hölder continuous mappings on Heisenberg groups.
method Theory of distributional Jacobians and pullbacks of differential forms.
result Simple proof of a generalization of the Gromov non-embedding theorem and new results about Hölder homotopy groups.
Study geodesics in Heisenberg groups with sub-Finsler metrics.
problem Characterize infinite geodesics in Heisenberg groups.
method Optimal control theory applied to sub-Finsler metrics.
result Proves infinite geodesics are horizontal lines under specific conditions.
In this paper we study the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group. We prove that all of the non-geodesic non-null biharmonic curves in 3-dimensional hyperbolic Heisenberg group are helices. Moreover, we obtain explicit parametric equations for non-geodesic non-null biharmon…
Study on Killing magnetic curves in Heisenberg group geometry.
problem Understanding Killing magnetic curves in Heisenberg group.
method Presentation of Heisenberg group geometry and geodesics, study of Killing magnetic curves with explicit formulas.
result Explicit formulas for Killing magnetic curves in Heisenberg group.
Sharp stability of isometries on Heisenberg group proven.
problem Quantitative stability of isometries on the Heisenberg group.
method Proving quasi-isometries close to isometries with specific closeness orders.
result Quasi-isometries of Heisenberg group are close to isometries with specific closeness orders.
Study local control in a 7D quaternionic Heisenberg group.
problem Optimizing geodesics in a 7D quaternionic Heisenberg group.
method Matrix representation and analysis of sub-Riemannian structure symmetries.
result Impact of symmetries on geodesic optimality.